Find a basis for the plane $x-2y+3z=0$ in $R^3$. Then find a basis for the intersection of that plane with the $xy$ plane.

Is there a proper/algebraic way of finding the basis of a plane?
Just by looking at it a basis could be $(2, 1, 0)$ because any multiple of that will give you $0$ when you substitute, but how do I find this without guessing?

would I use the same process when finding the basis of a line?

Any hints on how to figure out the second part of the question?

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    $\begingroup$ A plane through the origin is a 2D subspace, so needs 2 basis vectors. The intersection of the plane with the $xy$ plane will be a 1D subspace, with 1 basis vector. $\endgroup$ – David Mar 18 '16 at 1:06
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    $\begingroup$ You can read off the normal vector of your plane. It is $(1,-2,3)$. Now, find the space of all vectors that are orthogonal to this vector (which then is the plane itself) and choose a basis from it. OR (easier): put in any 2 values for x and y and solve for z. Then $(x,y,z)$ is a point on the plane. Do that again with another random sample. Then the two choices are linearly independent with high probability. Check it and you're done. $\endgroup$ – Friedrich Philipp Mar 18 '16 at 1:12
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    $\begingroup$ You can use the idea in this question. In your case you have only one equation which means you have two free variables. $\endgroup$ – Mhenni Benghorbal Mar 18 '16 at 1:28

One thing you can identify is that $z = \frac{2y-x}{3}$, then the points that are going to satisfy the equality will be of the form $$ \left(x, y, \frac{2y-x}{3}\right). $$ For you to be able to cover all of such points, you would need to have two different vectors satisfying above such that they are not a multiple of each other.

You can see that indeed we can decompose the above vector as $$ \left(x, y, \frac{2y-x}{3}\right)=x\left(1,0,-\frac{1}{3}\right)+y\left(0,1,\frac{2}{3}\right). $$ which gives you an obvious basis $$ (v_1,v_2)=\left(1,0,-\frac{1}{3}\right)+\left(0,1,\frac{2}{3}\right). $$ There are many different ways of constructing a vector given in the first characterization, which would result in how the basis vector are aligned with respect to each other.


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